$□φ = f$

Let $V = \sqrt{|g|}$

$\frac{1}{V} (V g^{ij} φ_{,i})_{,j} = f$

...as finite difference...

$\frac{1}{V} D_j (V g^{ij} D_i φ) = f$

expanding the inner finite difference term:

$\frac{1}{V[t,x,y,z]} ($
$ D_j( V[t,x,y,z] ($
$ \frac{1}{2 Δt} g^{tj}[t,x,y,z] (φ[t+1,x,y,z] - φ[t-1,x,y,z])$
$ + \frac{1}{2 Δx} g^{xj}[t,x,y,z] (φ[t,x+1,y,z] - φ[t,x-1,y,z])$
$ + \frac{1}{2 Δy} g^{yj}[t,x,y,z] (φ[t,x,y+1,z] - φ[t,x,y-1,z])$
$ + \frac{1}{2 Δz} g^{zj}[t,x,y,z] (φ[t,x,y,z+1] - φ[t,x,y,z-1])$
$ ))$
$) = f$

expanding the outer finite difference terms:

$\frac{1}{V[t,x,y,z]} ($
$ \frac{1}{2 Δt} ($
$ V[t+1,x,y,z] ($
$ \frac{1}{2 Δt} g^{tt}[t+1,x,y,z] (φ[t+2,x,y,z] - φ[t,x,y,z])$
$ + \frac{1}{2 Δx} g^{xt}[t+1,x,y,z] (φ[t+1,x+1,y,z] - φ[t+1,x-1,y,z])$
$ + \frac{1}{2 Δy} g^{yt}[t+1,x,y,z] (φ[t+1,x,y+1,z] - φ[t+1,x,y-1,z])$
$ + \frac{1}{2 Δz} g^{zt}[t+1,x,y,z] (φ[t+1,x,y,z+1] - φ[t+1,x,y,z-1])$
$ ) - V[t-1,x,y,z] ($
$ \frac{1}{2 Δt} g^{tt}[t-1,x,y,z] (φ[t,x,y,z] - φ[t-2,x,y,z])$
$ + \frac{1}{2 Δx} g^{xt}[t-1,x,y,z] (φ[t-1,x+1,y,z] - φ[t-1,x-1,y,z])$
$ + \frac{1}{2 Δy} g^{yt}[t-1,x,y,z] (φ[t-1,x,y+1,z] - φ[t-1,x,y-1,z])$
$ + \frac{1}{2 Δz} g^{zt}[t-1,x,y,z] (φ[t-1,x,y,z+1] - φ[t-1,x,y,z-1])$
$ )$
$ )$
$ + \frac{1}{2 Δx} ($
$ V[t,x+1,y,z] ($
$ \frac{1}{2 Δt} g^{tx}[t,x+1,y,z] (φ[t+1,x+1,y,z] - φ[t-1,x+1,y,z])$
$ + \frac{1}{2 Δx} g^{xx}[t,x+1,y,z] (φ[t,x+2,y,z] - φ[t,x,y,z])$
$ + \frac{1}{2 Δy} g^{yx}[t,x+1,y,z] (φ[t,x+1,y+1,z] - φ[t,x+1,y-1,z])$
$ + \frac{1}{2 Δz} g^{zx}[t,x+1,y,z] (φ[t,x+1,y,z+1] - φ[t,x+1,y,z-1])$
$ ) - V[t,x-1,y,z] ($
$ \frac{1}{2 Δt} g^{tx}[t,x-1,y,z] (φ[t+1,x-1,y,z] - φ[t-1,x-1,y,z])$
$ + \frac{1}{2 Δx} g^{xx}[t,x-1,y,z] (φ[t,x,y,z] - φ[t,x-2,y,z])$
$ + \frac{1}{2 Δy} g^{yx}[t,x-1,y,z] (φ[t,x-1,y+1,z] - φ[t,x-1,y-1,z])$
$ + \frac{1}{2 Δz} g^{zx}[t,x-1,y,z] (φ[t,x-1,y,z+1] - φ[t,x-1,y,z-1])$
$ )$
$ )$
$ + \frac{1}{2 Δy} ($
$ V[t,x,y+1,z] ($
$ \frac{1}{2 Δt} g^{ty}[t,x,y+1,z] (φ[t+1,x,y+1,z] - φ[t-1,x,y+1,z])$
$ + \frac{1}{2 Δx} g^{xy}[t,x,y+1,z] (φ[t,x+1,y+1,z] - φ[t,x-1,y+1,z])$
$ + \frac{1}{2 Δy} g^{yy}[t,x,y+1,z] (φ[t,x,y+2,z] - φ[t,x,y,z])$
$ + \frac{1}{2 Δz} g^{zy}[t,x,y+1,z] (φ[t,x,y+1,z+1] - φ[t,x,y+1,z-1])$
$ ) - V[t,x,y-1,z] ($
$ \frac{1}{2 Δt} g^{ty}[t,x,y-1,z] (φ[t+1,x,y-1,z] - φ[t-1,x,y-1,z])$
$ + \frac{1}{2 Δx} g^{xy}[t,x,y-1,z] (φ[t,x+1,y-1,z] - φ[t,x-1,y-1,z])$
$ + \frac{1}{2 Δy} g^{yy}[t,x,y-1,z] (φ[t,x,y,z] - φ[t,x,y-2,z])$
$ + \frac{1}{2 Δz} g^{zy}[t,x,y-1,z] (φ[t,x,y-1,z+1] - φ[t,x,y-1,z-1])$
$ )$
$ )$
$ + \frac{1}{2 Δz} ($
$ V[t,x,y,z+1] ($
$ \frac{1}{2 Δt} g^{tz}[t,x,y,z+1] (φ[t+1,x,y,z+1] - φ[t-1,x,y,z+1])$
$ + \frac{1}{2 Δx} g^{xz}[t,x,y,z+1] (φ[t,x+1,y,z+1] - φ[t,x-1,y,z+1])$
$ + \frac{1}{2 Δy} g^{yz}[t,x,y,z+1] (φ[t,x,y+1,z+1] - φ[t,x,y-1,z+1])$
$ + \frac{1}{2 Δz} g^{zz}[t,x,y,z+1] (φ[t,x,y,z+2] - φ[t,x,y,z])$
$ ) - V[t,x,y,z-1] ($
$ \frac{1}{2 Δt} g^{tz}[t,x,y,z-1] (φ[t+1,x,y,z-1] - φ[t-1,x,y,z-1])$
$ + \frac{1}{2 Δx} g^{xz}[t,x,y,z-1] (φ[t,x+1,y,z-1] - φ[t,x-1,y,z-1])$
$ + \frac{1}{2 Δy} g^{yz}[t,x,y,z-1] (φ[t,x,y+1,z-1] - φ[t,x,y-1,z-1])$
$ + \frac{1}{2 Δz} g^{zz}[t,x,y,z-1] (φ[t,x,y,z] - φ[t,x,y,z-2])$
$ )$
$ )$
$) = f$

solve for φ[t+2,x,y,z]

$φ[t+2,x,y,z] $
$= φ[t,x,y,z]$
$- Δt \frac{1}{g^{tt}[t+1,x,y,z]} ($
$ \frac{1}{Δx} g^{tx}[t+1,x,y,z] (φ[t+1,x+1,y,z] - φ[t+1,x-1,y,z])$
$ + \frac{1}{Δy} g^{ty}[t+1,x,y,z] (φ[t+1,x,y+1,z] - φ[t+1,x,y-1,z])$
$ + \frac{1}{Δz} g^{tz}[t+1,x,y,z] (φ[t+1,x,y,z+1] - φ[t+1,x,y,z-1])$
$ - \frac{V[t-1,x,y,z]}{V[t+1,x,y,z]} ($
$ \frac{1}{Δt} g^{tt}[t-1,x,y,z] (φ[t,x,y,z] - φ[t-2,x,y,z])$
$ + \frac{1}{Δx} g^{tx}[t-1,x,y,z] (φ[t-1,x+1,y,z] - φ[t-1,x-1,y,z])$
$ + \frac{1}{Δy} g^{ty}[t-1,x,y,z] (φ[t-1,x,y+1,z] - φ[t-1,x,y-1,z])$
$ + \frac{1}{Δz} g^{tz}[t-1,x,y,z] (φ[t-1,x,y,z+1] - φ[t-1,x,y,z-1])$
$ )$
$) $
$- \frac{Δt^2}{V[t+1,x,y,z] g^{tt}[t+1,x,y,z]} ($
$ \frac{1}{Δx} ($
$ V[t,x+1,y,z] ($
$ \frac{1}{Δt} g^{tx}[t,x+1,y,z] (φ[t+1,x+1,y,z] - φ[t-1,x+1,y,z])$
$ + \frac{1}{Δx} g^{xx}[t,x+1,y,z] (φ[t,x+2,y,z] - φ[t,x,y,z])$
$ + \frac{1}{Δy} g^{xy}[t,x+1,y,z] (φ[t,x+1,y+1,z] - φ[t,x+1,y-1,z])$
$ + \frac{1}{Δz} g^{xz}[t,x+1,y,z] (φ[t,x+1,y,z+1] - φ[t,x+1,y,z-1])$
$ ) - V[t,x-1,y,z] ($
$ \frac{1}{Δt} g^{tx}[t,x-1,y,z] (φ[t+1,x-1,y,z] - φ[t-1,x-1,y,z])$
$ + \frac{1}{Δx} g^{xx}[t,x-1,y,z] (φ[t,x,y,z] - φ[t,x-2,y,z])$
$ + \frac{1}{Δy} g^{xy}[t,x-1,y,z] (φ[t,x-1,y+1,z] - φ[t,x-1,y-1,z])$
$ + \frac{1}{Δz} g^{xz}[t,x-1,y,z] (φ[t,x-1,y,z+1] - φ[t,x-1,y,z-1])$
$ )$
$ )$
$ + \frac{1}{Δy} ($
$ V[t,x,y+1,z] ($
$ \frac{1}{Δt} g^{ty}[t,x,y+1,z] (φ[t+1,x,y+1,z] - φ[t-1,x,y+1,z])$
$ + \frac{1}{Δx} g^{xy}[t,x,y+1,z] (φ[t,x+1,y+1,z] - φ[t,x-1,y+1,z])$
$ + \frac{1}{Δy} g^{yy}[t,x,y+1,z] (φ[t,x,y+2,z] - φ[t,x,y,z])$
$ + \frac{1}{Δz} g^{yz}[t,x,y+1,z] (φ[t,x,y+1,z+1] - φ[t,x,y+1,z-1])$
$ ) - V[t,x,y-1,z] ($
$ \frac{1}{Δt} g^{ty}[t,x,y-1,z] (φ[t+1,x,y-1,z] - φ[t-1,x,y-1,z])$
$ + \frac{1}{Δx} g^{xy}[t,x,y-1,z] (φ[t,x+1,y-1,z] - φ[t,x-1,y-1,z])$
$ + \frac{1}{Δy} g^{yy}[t,x,y-1,z] (φ[t,x,y,z] - φ[t,x,y-2,z])$
$ + \frac{1}{Δz} g^{yz}[t,x,y-1,z] (φ[t,x,y-1,z+1] - φ[t,x,y-1,z-1])$
$ )$
$ )$
$ + \frac{1}{Δz} ($
$ V[t,x,y,z+1] ($
$ \frac{1}{Δt} g^{tz}[t,x,y,z+1] (φ[t+1,x,y,z+1] - φ[t-1,x,y,z+1])$
$ + \frac{1}{Δx} g^{xz}[t,x,y,z+1] (φ[t,x+1,y,z+1] - φ[t,x-1,y,z+1])$
$ + \frac{1}{Δy} g^{yz}[t,x,y,z+1] (φ[t,x,y+1,z+1] - φ[t,x,y-1,z+1])$
$ + \frac{1}{Δz} g^{zz}[t,x,y,z+1] (φ[t,x,y,z+2] - φ[t,x,y,z])$
$ ) - V[t,x,y,z-1] ($
$ \frac{1}{Δt} g^{tz}[t,x,y,z-1] (φ[t+1,x,y,z-1] - φ[t-1,x,y,z-1])$
$ + \frac{1}{Δx} g^{xz}[t,x,y,z-1] (φ[t,x+1,y,z-1] - φ[t,x-1,y,z-1])$
$ + \frac{1}{Δy} g^{yz}[t,x,y,z-1] (φ[t,x,y+1,z-1] - φ[t,x,y-1,z-1])$
$ + \frac{1}{Δz} g^{zz}[t,x,y,z-1] (φ[t,x,y,z] - φ[t,x,y,z-2])$
$ )$
$ )$
$ + \frac{1}{V[t,x,y,z]} f$
$)$

substitute 2012 Visser acoustic black hole metric:
$g^{tt} = -\frac{1}{c^2}$
$g^{ti} = -\frac{v^i}{c^2}$
$g^{ij} = \delta^{ij} - \frac{v^i v^j}{c^2}$

$φ[t+2,x,y,z] $
$= φ[t,x,y,z]$
$+ Δt ($
$ \frac{1}{Δt} \frac{V[t-1,x,y,z]}{V[t+1,x,y,z]} (φ[t,x,y,z] - φ[t-2,x,y,z])$
$ + \frac{1}{Δx} ($
$ \beta^x[t+1,x,y,z] (φ[t+1,x+1,y,z] - φ[t+1,x-1,y,z])$
$ - \beta^x[t-1,x,y,z] (φ[t-1,x+1,y,z] - φ[t-1,x-1,y,z]) \frac{V[t-1,x,y,z]}{V[t+1,x,y,z]}$
$ )$
$ + \frac{1}{Δy} ($
$ \beta^y[t+1,x,y,z] (φ[t+1,x,y+1,z] - φ[t+1,x,y-1,z])$
$ - \beta^y[t-1,x,y,z] (φ[t-1,x,y+1,z] - φ[t-1,x,y-1,z]) \frac{V[t-1,x,y,z]}{V[t+1,x,y,z]}$
$ )$
$ + \frac{1}{Δz} ($
$ \beta^z[t+1,x,y,z] (φ[t+1,x,y,z+1] - φ[t+1,x,y,z-1])$
$ - \beta^z[t-1,x,y,z] (φ[t-1,x,y,z+1] - φ[t-1,x,y,z-1]) \frac{V[t-1,x,y,z]}{V[t+1,x,y,z]}$
$ )$
$ + \frac{Δt}{V[t+1,x,y,z]} (c[x,y,z])^2 ($
$ \frac{1}{Δx} ($
$ V[t,x+1,y,z] ($
$ \frac{1}{Δt} ( \frac{\beta^x}{c^2} )[t,x+1,y,z] (φ[t+1,x+1,y,z] - φ[t-1,x+1,y,z])$
$ + \frac{1}{Δx} g^{xx}[t,x+1,y,z] (φ[t,x+2,y,z] - φ[t,x,y,z])$
$ + \frac{1}{Δy} g^{xy}[t,x+1,y,z] (φ[t,x+1,y+1,z] - φ[t,x+1,y-1,z])$
$ + \frac{1}{Δz} g^{xz}[t,x+1,y,z] (φ[t,x+1,y,z+1] - φ[t,x+1,y,z-1])$
$ ) - V[t,x-1,y,z] ($
$ \frac{1}{Δt} ( \frac{\beta^x}{c^2} )[t,x-1,y,z] (φ[t+1,x-1,y,z] - φ[t-1,x-1,y,z])$
$ + \frac{1}{Δx} g^{xx}[t,x-1,y,z] (φ[t,x,y,z] - φ[t,x-2,y,z])$
$ + \frac{1}{Δy} g^{xy}[t,x-1,y,z] (φ[t,x-1,y+1,z] - φ[t,x-1,y-1,z])$
$ + \frac{1}{Δz} g^{xz}[t,x-1,y,z] (φ[t,x-1,y,z+1] - φ[t,x-1,y,z-1])$
$ )$
$ )$
$ + \frac{1}{Δy} ($
$ V[t,x,y+1,z] ($
$ \frac{1}{Δt} ( \frac{\beta^y}{c^2} )[t,x,y+1,z] (φ[t+1,x,y+1,z] - φ[t-1,x,y+1,z])$
$ + \frac{1}{Δx} g^{xy}[t,x,y+1,z] (φ[t,x+1,y+1,z] - φ[t,x-1,y+1,z])$
$ + \frac{1}{Δy} g^{yy}[t,x,y+1,z] (φ[t,x,y+2,z] - φ[t,x,y,z])$
$ + \frac{1}{Δz} g^{yz}[t,x,y+1,z] (φ[t,x,y+1,z+1] - φ[t,x,y+1,z-1])$
$ ) - V[t,x,y-1,z] ($
$ \frac{1}{Δt} ( \frac{\beta^y}{c^2} )[t,x,y-1,z] (φ[t+1,x,y-1,z] - φ[t-1,x,y-1,z])$
$ + \frac{1}{Δx} g^{xy}[t,x,y-1,z] (φ[t,x+1,y-1,z] - φ[t,x-1,y-1,z])$
$ + \frac{1}{Δy} g^{yy}[t,x,y-1,z] (φ[t,x,y,z] - φ[t,x,y-2,z])$
$ + \frac{1}{Δz} g^{yz}[t,x,y-1,z] (φ[t,x,y-1,z+1] - φ[t,x,y-1,z-1])$
$ )$
$ )$
$ + \frac{1}{Δz} ($
$ V[t,x,y,z+1] ($
$ \frac{1}{Δt} ( \frac{\beta^z}{c^2} )[t,x,y,z+1] (φ[t+1,x,y,z+1] - φ[t-1,x,y,z+1])$
$ + \frac{1}{Δx} g^{xz}[t,x,y,z+1] (φ[t,x+1,y,z+1] - φ[t,x-1,y,z+1])$
$ + \frac{1}{Δy} g^{yz}[t,x,y,z+1] (φ[t,x,y+1,z+1] - φ[t,x,y-1,z+1])$
$ + \frac{1}{Δz} g^{zz}[t,x,y,z+1] (φ[t,x,y,z+2] - φ[t,x,y,z])$
$ ) - V[t,x,y,z-1] ($
$ \frac{1}{Δt} ( \frac{\beta^z}{c^2} )[t,x,y,z-1] (φ[t+1,x,y,z-1] - φ[t-1,x,y,z-1])$
$ + \frac{1}{Δx} g^{xz}[t,x,y,z-1] (φ[t,x+1,y,z-1] - φ[t,x-1,y,z-1])$
$ + \frac{1}{Δy} g^{yz}[t,x,y,z-1] (φ[t,x,y+1,z-1] - φ[t,x,y-1,z-1])$
$ + \frac{1}{Δz} g^{zz}[t,x,y,z-1] (φ[t,x,y,z] - φ[t,x,y,z-2])$
$ )$
$ )$
$ + \frac{1}{V[t,x,y,z]} f$
$ )$
$)$